eja: factor out a class for real-embedded matrices.

[sage.d.git] / mjo / eja / eja_utils.py

diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py
index 38e75761dab0394f3aa5e6e3016aed7c0edebbc8..2402d9f0e5f6d1b174690688b8f9b2d7f17eba61 100644 (file)
--- a/mjo/eja/eja_utils.py
+++ b/mjo/eja/eja_utils.py
@@ -2,14 +2,67 @@ from sage.functions.other import sqrt
 from sage.matrix.constructor import matrix
 from sage.modules.free_module_element import vector
 
+def _scale(x, alpha):
+    r"""
+    Scale the vector, matrix, or cartesian-product-of-those-things
+    ``x`` by ``alpha``.
+
+    This works around the inability to scale certain elements of
+    Cartesian product spaces, as reported in
+
+      https://trac.sagemath.org/ticket/31435
+
+    ..WARNING:
+
+        This will do the wrong thing if you feed it a tuple or list.
+
+    SETUP::
+
+        sage: from mjo.eja.eja_utils import _scale
+
+    EXAMPLES::
+
+        sage: v = vector(QQ, (1,2,3))
+        sage: _scale(v,2)
+        (2, 4, 6)
+        sage: m = matrix(QQ, [[1,2],[3,4]])
+        sage: M = cartesian_product([m.parent(), m.parent()])
+        sage: _scale(M((m,m)), 2)
+        ([2 4]
+        [6 8], [2 4]
+        [6 8])
+
+    """
+    if hasattr(x, 'cartesian_factors'):
+        P = x.parent()
+        return P(tuple( _scale(x_i, alpha)
+                        for x_i in x.cartesian_factors() ))
+    else:
+        return x*alpha
+
+
 def _all2list(x):
     r"""
     Flatten a vector, matrix, or cartesian product of those things
     into a long list.
 
-    EXAMPLES::
+    If the entries of the matrix themselves belong to a real vector
+    space (such as the complex numbers which can be thought of as
+    pairs of real numbers), they will also be expanded in vector form
+    and flattened into the list.
+
+    SETUP::
 
         sage: from mjo.eja.eja_utils import _all2list
+        sage: from mjo.octonions import Octonions, OctonionMatrixAlgebra
+
+    EXAMPLES::
+
+        sage: _all2list([[1]])
+        [1]
+
+    ::
+
         sage: V1 = VectorSpace(QQ,2)
         sage: V2 = MatrixSpace(QQ,2)
         sage: x1 = V1([1,1])
@@ -26,15 +79,45 @@ def _all2list(x):
         sage: _all2list(M((x2,y2)))
         [1, -1, 0, 1, 1, 0]
 
+    ::
+
+        sage: _all2list(Octonions().one())
+        [1, 0, 0, 0, 0, 0, 0, 0]
+        sage: _all2list(OctonionMatrixAlgebra(1).one())
+        [1, 0, 0, 0, 0, 0, 0, 0]
+
+    ::
+
+        sage: V1 = VectorSpace(QQ,2)
+        sage: V2 = OctonionMatrixAlgebra(1,field=QQ)
+        sage: C = cartesian_product([V1,V2])
+        sage: x1 = V1([3,4])
+        sage: y1 = V2.one()
+        sage: _all2list(C( (x1,y1) ))
+        [3, 4, 1, 0, 0, 0, 0, 0, 0, 0]
+
     """
-    if hasattr(x, 'list'):
-        # Easy case...
-        return x.list()
-    else:
-        # But what if it's a tuple or something else? This has to
-        # handle cartesian products of cartesian products, too; that's
-        # why it's recursive.
-        return sum( map(_all2list,x), [] )
+    if hasattr(x, 'to_vector'):
+        # This works on matrices of e.g. octonions directly, without
+        # first needing to convert them to a list of octonions and
+        # then recursing down into the list. It also avoids the wonky
+        # list(x) when x is an element of a CFM. I don't know what it
+        # returns but it aint the coordinates. This will fall through
+        # to the iterable case the next time around.
+        return _all2list(x.to_vector())
+
+    try:
+        xl = list(x)
+    except TypeError: # x is not iterable
+        return [x]
+
+    if xl == [x]:
+        # Avoid the retardation of list(QQ(1)) == [1].
+        return [x]
+
+    return sum(list( map(_all2list, xl) ), [])
+
+
 
 def _mat2vec(m):
         return vector(m.base_ring(), m.list())
@@ -160,18 +243,16 @@ def gram_schmidt(v, inner_product=None):
 
     R = v[0].base_ring()
 
-    # Define a scaling operation that can be used on tuples.
-    # Oh and our "zero" needs to belong to the right space.
-    scale = lambda x,alpha: x*alpha
+    # Our "zero" needs to belong to the right space for sum() to work.
     zero = v[0].parent().zero()
-    if hasattr(v[0], 'cartesian_factors'):
-        P = v[0].parent()
-        scale = lambda x,alpha: P(tuple( x_i*alpha
-                                         for x_i in x.cartesian_factors() ))
 
+    sc = lambda x,a: a*x
+    if hasattr(v[0], 'cartesian_factors'):
+        # Only use the slow implementation if necessary.
+        sc = _scale
 
     def proj(x,y):
-        return scale(x, (inner_product(x,y)/inner_product(x,x)))
+        return sc(x, (inner_product(x,y)/inner_product(x,x)))
 
     # First orthogonalize...
     for i in range(1,len(v)):
@@ -188,6 +269,6 @@ def gram_schmidt(v, inner_product=None):
     # them here because then our subalgebra would have a bigger field
     # than the superalgebra.
     for i in range(len(v)):
-        v[i] = scale(v[i], ~norm(v[i]))
+        v[i] = sc(v[i], ~norm(v[i]))
 
     return v